Spectral doubling of normal operators and connections with antiunitary operators (Q421099)

Let \(H\) be a complex separable Hilbert space and let \(B(H)\) denote the algebra of all bounded linear operators on \(H\). An antilinear map \(J:H\to H\) is called antiunitary if it is onto and \((Jx,Jy)=(y,x)\) for every \(x,y\in H\). If, in addition, \(J^2=I\), then \(J\) is called a conjugation. The main result of this interesting paper is the following Theorem. Suppose that \(A,B\in B(H)\) are normal operators with \(A\) invertible, and \(J\) is a conjugation such that \(AB=(JB^*J)A\). Then the restriction of \(B\) to \((\text{Ker}(A-JA^*J))^\perp\) has even multiplicity. A nice corollary is the following. If \(B\in B(H)\) is a normal operator with simple spectrum and \(K\) is antiunitary with \(B^*K=KB\), then \(K\) is a conjugation. These results are then used to study generalized notions of Hamiltonian operators.